I recently used ubuntu 12.10 and linux mint 14 and found that ubuntu has good gui interface while booting and shutdown like windows. Is there any way I change my startup and shutdown gui in linuxmint. I don't like the CLI black screen. And recommend this gui interface prebuilt in linux mint 15.

You mean the login window? Linux Mint uses its own called MDM, which is starting to look awesome for Linux Mint 15 as you can theme the login window with HTML and Javascript. See samriggs' awesome examples here: viewtopic.php?f=25&t=126979. Ubuntu uses LightDM for the login window.

Splash screens shown while booting (until the login window appears) are done by Plymouth. Both on Linux Mint and Ubuntu. You can use a different Plymouth theme if you want. There are a couple of topics for that already IIRC.